Properties

Label 416.2.f.d
Level $416$
Weight $2$
Character orbit 416.f
Analytic conductor $3.322$
Analytic rank $0$
Dimension $4$
CM discriminant -52
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [416,2,Mod(129,416)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(416, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("416.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 416.f (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.32177672409\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{7} - 3 q^{9} + (\beta_{2} + \beta_1) q^{11} - \beta_{3} q^{13} + 2 \beta_{3} q^{17} + ( - \beta_{2} + \beta_1) q^{19} + 5 q^{25} - 2 \beta_{3} q^{29} + (2 \beta_{2} + \beta_1) q^{31} + ( - 2 \beta_{2} + \beta_1) q^{47} + (2 \beta_{3} - 7) q^{49} - 2 q^{53} + ( - \beta_{2} - 3 \beta_1) q^{59} + 6 q^{61} - 3 \beta_1 q^{63} + (3 \beta_{2} + \beta_1) q^{67} + ( - 2 \beta_{2} - 3 \beta_1) q^{71} + ( - 2 \beta_{3} - 10) q^{77} + 9 q^{81} + (\beta_{2} - 3 \beta_1) q^{83} + ( - 3 \beta_{2} + \beta_1) q^{91} + ( - 3 \beta_{2} - 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 20 q^{25} - 28 q^{49} - 8 q^{53} + 24 q^{61} - 40 q^{77} + 36 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{3} + 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{2} - 8\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
2.30278i
1.30278i
1.30278i
2.30278i
0 0 0 0 0 4.60555i 0 −3.00000 0
129.2 0 0 0 0 0 2.60555i 0 −3.00000 0
129.3 0 0 0 0 0 2.60555i 0 −3.00000 0
129.4 0 0 0 0 0 4.60555i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.b odd 2 1 CM by \(\Q(\sqrt{-13}) \)
4.b odd 2 1 inner
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 416.2.f.d 4
3.b odd 2 1 3744.2.c.g 4
4.b odd 2 1 inner 416.2.f.d 4
8.b even 2 1 832.2.f.g 4
8.d odd 2 1 832.2.f.g 4
12.b even 2 1 3744.2.c.g 4
13.b even 2 1 inner 416.2.f.d 4
13.d odd 4 1 5408.2.a.v 2
13.d odd 4 1 5408.2.a.y 2
39.d odd 2 1 3744.2.c.g 4
52.b odd 2 1 CM 416.2.f.d 4
52.f even 4 1 5408.2.a.v 2
52.f even 4 1 5408.2.a.y 2
104.e even 2 1 832.2.f.g 4
104.h odd 2 1 832.2.f.g 4
156.h even 2 1 3744.2.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
416.2.f.d 4 1.a even 1 1 trivial
416.2.f.d 4 4.b odd 2 1 inner
416.2.f.d 4 13.b even 2 1 inner
416.2.f.d 4 52.b odd 2 1 CM
832.2.f.g 4 8.b even 2 1
832.2.f.g 4 8.d odd 2 1
832.2.f.g 4 104.e even 2 1
832.2.f.g 4 104.h odd 2 1
3744.2.c.g 4 3.b odd 2 1
3744.2.c.g 4 12.b even 2 1
3744.2.c.g 4 39.d odd 2 1
3744.2.c.g 4 156.h even 2 1
5408.2.a.v 2 13.d odd 4 1
5408.2.a.v 2 52.f even 4 1
5408.2.a.y 2 13.d odd 4 1
5408.2.a.y 2 52.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(416, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 28T^{2} + 144 \) Copy content Toggle raw display
$11$ \( T^{4} + 44T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 76T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 52)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 124T^{2} + 1296 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 188T^{2} + 4624 \) Copy content Toggle raw display
$53$ \( (T + 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 236 T^{2} + 13456 \) Copy content Toggle raw display
$61$ \( (T - 6)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 268 T^{2} + 11664 \) Copy content Toggle raw display
$71$ \( T^{4} + 284T^{2} + 8464 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 332T^{2} + 4624 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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